Simulative analyse of turbine trains under blade fracture conditions with regard to

the implementation methods of journal

bearings

E. Woschke, C. Daniel, S. Nitzschke

Otto-von-Guericke-University Magdeburg, Faculty of Mechanical Engineering, Insti-

tute of Mechanics (Fluid Structure Interaction & Technical Dynamics), Germany

ABSTRACT

Concerning the design of rotor systems supported by journal bearings, worst-case

scenarios represent a highly sensitive investigation. Beside the actual rotor dynamic

parameters, especially the nonlinear bearing characteristics have a significant influ-

ence on the behaviour of the system and the resulting risk potential. The paper

gives an overview on the possibilities of numerical analysis of rotor systems with

journal bearings under dynamic conditions. Simulation results for a turbine train

excited by a sudden blade fracture are analysed using different modelling ap-

proaches concerning the bearings in context of reliability and simulation time.

1 INTRODUCTION

Due to the high effort of experimental investigations concerning dynamical load

cases, simulations are increasingly used. Depending on the actual application tran-

sient system behaviour refers to run-up or run-down situations, worst-case scenar-

ios or even typical operating conditions.

In conventional rotordynamic simulations the nonlinear behaviour of the journal

bearings is often implemented using significant simplifications. Although the basic

differential equation for the description of the hydrodynamic effects are known, a

holistic solution is numerical costly especially in context of time integration

schemes. As a result, often lookup tables are used ignoring the effects of misalign-

ments or even analytical formulations like short bearing theory without the possibil-

ity to consider boundary conditions for oil supply.

On the other hand, for simulations with focus on the hydrodynamic bearings the

rotor is often oversimplified to a mass point and the mechanical system behaviour

is modelled under steady-state conditions whereas the hydrodynamic equations are

solved with high modelling depth.

Especially for critical system conditions a reliable calculation of vibration ampli-

tudes, acting forces, minimal gaps etc. is crucial. At that point, advanced simula-

tions with both a detailed description of the hydrodynamics and an adequate mod-

elling of the rotordynamic behaviour are necessary.

2 MODEL OF THE TURBINE TRAIN

The turbine train to be investigated has a length of and shaft shoul-

ders varying in diameter from to . The turbine

wheels are modelled as rigid bodies with a mean diameter to achieve an adequate

rotor model by means of dominant stiffness and mass properties. A detailed de-

scription of the wheel-blades and their interaction is neither the focus of the paper

nor relevant for rotordynamic analyses.

Figure 1 - Modell of the turbine train including 11 journal bearings

The rotor is supported in journal bearings. The acting forces are determined by

gravity, leading to a static deformation of the shaft, and by unbalance at the tur-

bine wheels as well as on the generator. For general use, the rotor system will run

under steady-state conditions at a rotational speed of .

Beside linear investigations (e.g. eigenvalue calculation and harmonic analysis),

due to the behaviour of journal bearings, nonlinear rotordynamic simulations are

necessary, whereas the majority of computational methods is limited to the de-

scribed load case, with varying modelling depth concerning the hydrodynamics. In

case of dynamic conditions like due to a critical fault (e.g. blade fracture) the inves-

tigation of the rotor system becomes more sophisticated. For that task, a holistic

model including a direct interaction between rotordynamic and hydrodynamic is

needed.

3 ROTORDYNAMIC - EQUATIONS OF MOTION

The elastic properties of the turbine train are determined by the geometric dimen-

sions and the assigned material of the rotor. The assembly performs small motions,

except the rotation around the longitudinal axis. Therefore, under the restriction of

circular cross sections, the overall-motion can be described by means of a linear

finite element model. Also - beside the stiffness property - inertia and damping

effects have to be taken into account to be able to handle transient effects.

3.1 Finite beam element

As the main dimension of the shaft is determined by its length, considering the

existence of short shoulders, a beam model based on the Timoshenko-theory is

appropriate.

Following the approach for rotating structures given in (1), the equation of motions

becomes

Eq. 1

where denotes the mass matrix, the stiffness matrix and the skew-

symmetric gyroscopic matrix. Finally, the damping of the structure is considered by the Rayleigh approach .

The vector on the right-hand-side contains the distributed load due to gravity,

the unbalance induced forces and the reaction forces of the bearings, which are

described detailed in chapter 4.

journal bearings

LP turbine stages

generator

auxillary unit

HP turbine stage

support

Starting from some initial conditions for positions as well as for velocities , Eq. 1

is solved for the current accelerations , which are integrated numerically. Due to

the coupled hydro-structural dynamic task, a semi-implicit integration scheme,

which is capable to deal with stiff problems, is necessary.

The investigated turbine train consists of a variety of shaft shoulders, which pre-

supposes a fine discretisation and by that several degrees of freedom to ensure a

realistic modelling. However, since only some nodes are excited by external loads

or interact with the environment (nonlinear bearing forces), a system reduction is

suitable to reduce the computation time.

3.2 Improved reduction system method

For structural-mechanical simulations based on finite element formulations system

reduction is an effective way to reduce the numerical effort. Especially in the con-

text of dynamic simulations including non-linear force laws, which leads to numeri-

cal stiff problems, the number of degrees of freedom has to be restricted due to

demands of appropriate time integration methods.

Initially, a separation in master degrees of freedom (are still available after

reduction and can be loaded by forces and moments) and slave-degrees of freedom

(are no longer available after the reduction and has to be formally free of loads)

takes place. The application on the equation of motion yields the following form

Eq. 2

For the actual reduction a suitable transformation has to be applied in order to

describe the slave response as function of the master degrees of freedom

Eq. 3

Methods including a modal transformation (modal reduction, SEREP (2) etc.) are

not suitable for the reduction of rotordynamic structures due the dependency of

eigenvectors upon the angular velocity. However, there are different modal ap-

proaches but they are accompanied by a drastic numerical overhead and therefore

are of a more academic nature (3).

For this reason, more conventional reduction methods have to be used. To avoid

the known problems of static condensation (Guyan-reduction (4) - overestimation

of stiffness) and component mode synthesis (CMS-reduction (5) - problematic

choice of significant eigenmodes) a more generalised formulation called IRS-Method

(Improved Reduction System Method) is applied (6). In addition to the approach of Guyan given by

Eq. 4

pseudo-static forces are taken into account as a result of inertia terms (6)

Eq. 5

With the time derivation of the transformation in Eq. 4 the acceleration of the

slave-degrees of freedom can be formulated as

Eq. 6

Disregarding damping and gyroscopic terms, the equation of motion for the slave

part can be written as

Eq. 7

After a few rearrangements the transformation reads as

Eq. 8

An improvement of the transformation can be gained by repeatedly executing the

reduction. Though the stiffness and mass matrices of the static condensation have

to be replaced by the resulting matrices of the previous IRS-step

Eq. 9

where index designates the th iteration. The algorithm converges monotonically

with respect to the eigenvalues and eigenvectors of the unreduced system and by

that is suitable for dynamic simulations.

In Figure 2 a comparison between the eigenfrequency dependence upon the rota-

tional speed (Campbell-diagram) is presented for the examined turbine train under

free-free boundary conditions. The results show an appropriate accordance between

the dynamical behaviour of the reduced system (IRS) and the full model. Addition-

ally, commonly used reduction methods (Guyan and CMS) are also included. As

expected, the differences are higher w.r.t. the IRS reduction and increase with

rotational speed. To ensure a better coincidence for Guyan and CMS, additional

degrees of freedom has to be taken into account, whereas IRS method just needs

more iteration steps without any necessity to increase system dimensions.

Figure 2 - Campbell-diagram of the examined rotor for different reduction

methods

Finally, the reduced system matrices with can be written as

Eq. 10

so that the equation of motion becomes

Eq. 11

4 HYDRODYNAMIC - JOURNAL BEARINGS

Beside the mechanical model, also the nonlinear interaction of the bearings with the

rotor has to be described to allow for realistic system behaviour even under dynam-

ical conditions.

4.1 Online solution of Reynolds lubrication equation

Starting from conservation of mass and momentum for an infinitesimal volume

element the Navier-Stokes equations can be formulated. Further, by taking into

account geometrical dimensions and relations between velocities and fluid-flows in

different direction in the lubricating gap the Reynolds lubrication equation can be

derived

Eq. 12

where denotes the hydrodynamic pressure, the gap function, the rotational

speed and the radius of the shaft. Finally and are the relevant material pa-

rameters density and dynamic viscosity of the oil (7).

The solution of this partial elliptical differential equation is possible in a closed form

only for special cases of short and long bearings. Therefore a numerical method

(FDM, FVM, FEM) has to be used to enable the transformation into a system of

linear equations by discretisation of the equation in the solution area (8), (9).

By solving the system, negative pressure values appear formally, which can be

transmitted by fluids only to a very limited extend. Physically, the lubrication gap

diverges, on the one hand dissolved air in the fluid escapes and additional air is

drawn in via the axial boundaries whereas on the other hand the fluid vaporizes

(both phenomena are known as cavitation). Resulting, a two-phase flow of fluid and

compressible gas occurs. For modelling of the cavitation behaviour and in order to

achieve a physical correct solution, various methods differing in accuracy and nec-

essary computing time (JFO-theory (10), Elrod-algorithm (11), (12)) exist. Despite

this, with regard to the task of a dynamic simulation, the negative pressures are set

to zero (Gümbel cavitation algorithm) (13).

Figure 3 - Determination of gap function including misalignment

Usually, lookup tables for the bearing stiffness and damping properties based on

the linearized Reynolds equation are used. The nonlinear behaviour is modelled

exactly at the calculated states, deviations of these nominal states are determined

by interpolation (14). Problems arise especially when in addition to the pure trans-

lational movement of the shaft also rotations around the transverse axes occur,

e.g. due to unbalance response or bending of the shaft. Thus, the dimensionality of

the lookup tables rises and therefore the organisational effort increases drastically.

If these effects have a significant influence, the numerical solution of the Reynolds

lubrication equation is inevitable.

With focus on the dynamic behaviour of the rotor, the formulation of the gap func-

tion and its time derivation in each journal bearing including misalignment is

essential and can be accomplished as illustrated in Figure 3.

The gap function is derived by calculating the eccentricity in each axial position

of the bearing, which leads to

Eq. 13

The first derivate of the gap function has to be evaluated from the absolute ve-

locities of the shaft- and the housing surface, which are functions of the corre-sponding translational velocities and the angular velocities . Under usage

of the normal fractions and

Eq. 14

can be formulated by

Eq. 15

4.2 Scheme for holistic simulation

For the holistic simulation by means of a direct interaction between the mechanical

system and the hydrodynamic field problem of the journal bearings, an implemen-

tation of the nonlinear force laws in the dynamic simulation has to be realised (15).

Figure 4 - Scheme for holistic simulation of the rotor-bearing interaction

The top level instance is the time integration which solves the rotor system’s equa-

tion of motion in the state space. In every time step of the solver (including calcu-

lation of the jacobian etc.) the mechanical states are used to calculate the neces-

sary input quantities (gap-function and its time derivative) for the online solution of

the Reynolds lubrication equation. The resulting pressure is integrated to acting

forces and moments, which represents the nonlinear behaviour of the bearings.

Together with the outer forces due to gravity and unbalance the force vector can be

evaluated. As result of the acting forces the accelerations of the overall system can

be calculated and integrated in state space leading to the next time step (see Fig-

ure 4).

4.3 Modelling depth

For the further investigation three different modelling approaches V1-V3 are intro-

duced.

V1 full approach: The general implementation of the solution of the Reynolds differ-

ential equation - as described in chapter 4.1 - in the time integration allows a de-

tailed study of the dynamic system behaviour and the influence of individual pa-

rameters.

V2 look-up-table analogy:

Specifically, the misalignment in the bearings is neglected in the majority of studies

and simulation methods or is predefined by a value based on linear estimations.

These simplifications are in good agreement with those of popular solution using

lookup tables (16); hence this variant is of particular importance. In order to inves-

tigate the influence of these assumptions a force element was adapted, which ig-

nores the misalignment (see Figure 3) and solves the Reyn-

olds differential equation on the basis of displacement and velocity of the bearing

center.

V3 short bearing approximation:

Assuming that the bearing width is much smaller than the circumference (short

bearing) the Reynolds equation can be written as

Eq. 16

If further the misalignment can be neglected and the material parameters of the

fluid in the lubricating gap are constant , the variables can be sepa-

rated formally and the pressure distribution can be determined analytically

Eq. 17

where denote the ambient pressure. To minimize the numerical effort this

form is often preferred for dynamic analyses, although the results in the context of

the defined assumptions have to be scrutinized.

5 RESULTS

To analyse the dynamic system behaviour, at first steady-state conditions are mod-

elled with a nominal unbalance. To assure that all initial disturbances are vanished,

the system is simulated until at the unbalance suddenly increases to model

the blade fracture. As a result, the acting forces increase drastically as well as the

bearing forces and the vibration amplitudes. Figure 5 shows a scaled deformation of

the turbine train after the blade failure.

Figure 5 - Deformation and acting forces of turbine train under blade frac-

ture conditions

To determine the influence of the varying modelling depth for the journal bearings,

the essential output quantities are compared.

5.1 Steady state: unbalance response

Figure 7 shows the minimal gap in the analysed bearing. In the steady state region

until the minimal gap has a dominant constant part due

to the gravity force, which is superposed by small vibrations as

result of the unbalance. The mean value differs only slightly by

between the version with and without misalignment (which occurs due to unbalance

and bending), whereas the short bearing theory shows larger differences with re-

spect to the full model . As a result of the medium loading

(Sommerfeldnumber ) the bearing forces in Figure 6 show compa-

rable values in the steady state region.

Figure 6 - Maximum bearing forces and moments in the analysed bearing

depending on time

analysed bearing

Figure 7 - Minimal gap in the analysed bearing (top: full Reynolds with

misalignment, center: full Reynolds without misalignment, bot-

tom: short bearing theory)

Beside the hydrodynamic quantities, also the maximum vibration amplitudes pre-

sented in Figure 8 and the normal stresses in Figure 9 are relevant results for ro-

tordynamic respectively structural dynamic analyses. Again there are some minor

differences, but due to the moderate load and the steady state conditions the vibra-

tion amplitudes reach consistent values and the normal stresses are

similar for all modelling approaches .

From the steady state point of view all variants for the description of the nonlinear

behaviour of the system are suitable and there would be no need for an extended

simulation method including the numerical solution of the Reynolds lubrication

equation or the consideration of misalignment.

Figure 8 - Maximum vibration amplitudes in the shaft depending on time

Figure 9 - Maximum normal stress in the shaft depending on time

5.2 Transient: abrupt blade failure

While in case of steady-state conditions the results for the bearing quantities are in

good agreement, the transient analysis shows significant differences after the blade

loss. Figure 7 shows, that the minimal gap after the blade fracture at has

regions in which solid contact1 occurs . Due to the misalignment, the

minimal gap is not constantly at the same axial position - for sake of comparison

additionally the centerline minimal gap is plotted.

1 The contact is modelled using the Greenwood-Williamson-Tripp approach (17).

The model without misalignment is not able to show this behaviour, the lower min-

imal values and increased amplitudes here are a result of the higher unbalance

compared to the steady-state case. The differences are obvious, while model V1

predicts a critical failure, the results of the approach V2 would be tolerable. This

underlines the necessity to include misalignment in the description of the bearings.

The differences to the short bearing theory are even bigger. The simulated behav-

iour seems to be uncritical and there would be no reason to shut down the turbine

train, if only the approach V3 is available.

The high unbalance as a consequence of the blade loss causes the mean value of

the bearing forces to increase drastically ( ), Figure 6. Additionally,

due to the highly dynamical changes of contact condition the bearing forces are

fluctuating in the model V1 whereas the simplifications show an increased load level

but only slight dynamical influences and by that underestimate the maximum forces

by factor 3. Furthermore the bearing moments reaches significant values and

should not be ignored.

Taking a look at the maximum vibration amplitudes and maximum normal stresses

in Figure 8 and Figure 9 the differences as result of the bearing forces and the

bearing moment, which is only present under consideration of misalignment, are

obvious. By comparing these integral results the modelling depth leads to signifi-

cant discrepancies even between the simplified approaches V2 and V3, although the

trend is not clear (the maximum vibration amplitudes of V2 are lower than for the

approach V3 but for the maximum normal stresses this behaviour switches).

6 CONCLUSION

The paper at hand described a holistic simulation method for rotordynamic prob-

lems including a detailed description of the journal bearings, which is necessary due

to the significant influence on the system behaviour.

Especially for rotordynamic tasks and transient conditions the usual modelling

depth of the bearings is generally rather low. The presented results show a signifi-

cant influence of the bearing implementation upon tribological parameters and also

rotordynamical quantities. For steady-state conditions the modelling depth leads to

some minor differences, which after a sudden change of the unbalance resulting

from an assumed blade failure increase drastically. Whereas the full model deter-

mines solid contact in some bearings, the simplified models show a uncritical bear-

ing behaviour due to the neglected misalignment. As a result, the bearing forces

are underestimated and the vibration and stresses are calculated to low pretending

sufficient load capacities and a tolerable system behaviour.

Although the numerical effort for a holistic solution including a detailed description

of the bearings with misalignment effects is rather high, the results especially for

dynamical simulations and large displacements require such high modelling depth.

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